5 The Mueller Matrices for Polarizing Components 5.1 INTRODUCTION In the previous chapters we have concerned ourselves with the fundamental properties of polarized light In this chapter we now turn our attention to the study of the interaction of polarized light with elements which can change its state of polarization and see that the matrix representation of the Stokes parameters leads to a very powerful mathematical tool for treating this interaction In Fig 5-1 we show an incident beam interacting with a polarizing element and the emerging beam In Fig 5-1 the incident beam is characterized by its Stokes parameters Si, where i ¼ 0, 1, 2, The incident polarized beam interacts with the polarizing medium, and the emerging beam is characterized by a new set of Stokes parameters S 01 , where, again, i ¼ 0, 1, 2, We now assume that S 01 can be expressed as a linear combination of the four Stokes parameters of the incident beam by the relations: S 00 ¼ m00 S0 þ m01 S1 þ m02 S2 þ m03 S3 ð5-1aÞ S 01 ¼ m10 S0 þ m11 S1 þ m12 S2 þ m13 S3 ð5-1bÞ S 02 ¼ m20 S0 þ m21 S1 þ m22 S2 þ m23 S3 ð5-1cÞ S 03 ¼ m30 S0 þ m31 S1 þ m32 S2 þ m33 S3 ð5-1dÞ In matrix form (5-1) is 01 m00 S0 B S0 C B m B C B 10 B C¼B @ S A @ m20 S 03 m30 written as m01 m11 m02 m12 m21 m31 m22 m32 10 S0 m03 C B m13 CB S1 C C CB C m23 A@ S2 A m33 ð5-2Þ S3 or S0 ¼ M Á S Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-3Þ Figure 5-1 Interaction of a polarized beam with a polarizing element where S and S are the Stokes vectors and M is the  matrix known as the Mueller matrix It was introduced by Hans Mueller during the early 1940s While Mueller appears to have based his  matrix on a paper by F Perrin and a still earlier paper by P Soleillet, his name is firmly attached to it in the optical literature Mueller’s important contribution was that he, apparently, was the first to describe polarizing components in terms of his Mueller matrices Remarkably, Mueller never published his work on his matrices Their appearance in the optical literature was due to others, such as N.G Park III, a graduate student of Mueller’s who published Mueller’s ideas along with his own contributions and others shortly after the end of the Second World War When an optical beam interacts with matter its polarization state is almost always changed In fact, this appears to be the rule rather than the exception The polarization state can be changed by (1) changing the amplitudes, (2) changing the phase, (3) changing the direction of the orthogonal field components, or (4) transferring energy from polarized states to the unpolarized state An optical element that changes the orthogonal amplitudes unequally is called a polarizer or diattenuator Similarly, an optical device that introduces a phase shift between the orthogonal components is called a retarder; other names used for the same device are wave plate, compensator, or phase shifter If the optical device rotates the orthogonal components of the beam through an angle as it propagates through the element, it is called a rotator Finally, if energy in polarized states goes to the unpolarized state, the element is a depolarizer These effects are easily understood by writing the transverse field components for a plane wave: Ex ðz, tÞ ¼ E0x cosð!t À z þ x Þ ð5-4aÞ Ey ðz, tÞ ¼ E0y cosð!t À z þ y Þ ð5-4bÞ Equation (4) can be changed by varying the amplitudes, E0x or E0y, or the phase, x or y and, finally, the direction of Ex ðz, tÞ and Ey ðz, tÞ The corresponding devices for causing these changes are the polarizer, retarder, and rotator The use of Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved the names polarizer and retarder arose, historically, before the behavior of these polarizing elements was fully understood The preferable names would be diattenuator for a polarizer and phase shifter for the retarder All three polarizing elements, polarizer, retarder, and rotator, change the polarization state of an optical beam In the following sections we derive the Mueller matrices for these polarizing elements We then apply the Mueller matrix formalism to a number of problems of interest and see its great utility 5.2 THE MUELLER MATRIX OF A POLARIZER A polarizer is an optical element that attenuates the orthogonal components of an optical beam unequally; that is, a polarizer is an anisotropic attenuator; the two orthogonal transmission axes are designated px and py Recently, it has also been called a diattenuator, a more accurate and descriptive term A polarizer is sometimes described also by the terms generator and analyzer to refer to its use and position in the optical system If a polarizer is used to create polarized light, we call it a generator If it is used to analyze polarized light, it is called an analyzer If the orthogonal components of the incident beam are attenuated equally, then the polarizer becomes a neutral density filter We now derive the Mueller matrix for a polarizer In Fig 5-2 a polarized beam is shown incident on a polarizer along with the emerging beam The components of the incident beam are represented by Ex and Ey After the beam emerges from the polarizer the components are E 0x and E 0y , and they are parallel to the original axes The fields are related by E 0x ¼ px Ex px ð5-5aÞ E 0y ¼ py Ey py ð5-5bÞ The factors px and py are the amplitude attenuation coefficients along orthogonal transmission axes For no attenuation or perfect transmission along an orthogonal axis px ð py Þ ¼ 1, whereas for complete attenuation px ð py Þ ¼ If one of the axes has an absorption coefficient which is zero so that there is no transmission along this axis, the polarizer is said to have only a single transmission axis Figure 5-2 The Mueller matrix of a polarizer with attenuation coefficients px and py Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The Stokes polarization parameters of the incident and emerging beams are, respectively, S0 ¼ Ex Exà þ Ey Eyà ð5-6aÞ S1 ¼ Ex Exà À Ey Eyà ð5-6bÞ S2 ¼ Ex Eyà þ Ey Exà ð5-6cÞ S3 ¼ iðEx Eyà À Ey ExÃ Þ ð5-6dÞ 0à S 00 ¼ E 0x E 0à x þ EyEy ð5-7aÞ 0à S 01 ¼ E 0x E 0à x À EyEy ð5-7bÞ 0à S 02 ¼ E 0x E 0à y þ EyEx ð5-7cÞ 0à S 03 ¼ iðE 0x E 0à y À EyEx Þ ð5-7dÞ and Substituting (5-5) into (5-7) and using (5-6), we then find 01 10 px þ p2y p2x À p2y 0 S0 S0 B 0C B CB C 2 B S C B px À py px þ py BS C 0 C B C¼ B CB C B S0 C B CB S C 2px py A@ A @ 2A @ 0 S3 S3 0 2px py The  matrix in (5-8) is written by itself as px þ p2y p2x À p2y 0 B C B p2x À p2y p2x þ p2y 0 C C M¼ B C 2B 2px py A @ 0 0 px, y ð5-8Þ ð5-9Þ 2px py Equation (5-9) is the Mueller matrix for a polarizer with amplitude attenuation coefficients px and py In general, the existence of the m33 term shows that the polarization of the emerging beam of light will be elliptically polarized For a neutral density filter px ¼ py ¼ p and (5-9) becomes 1 0 B0 0C B C ð5-10Þ M ¼ p2 B C @0 0A 0 which is a unit diagonal matrix Equation (5-10) shows that the polarization state is not changed by a neutral density filter, but the intensity of the incident beam is reduced by a factor of p2 This is the expected behavior of a neutral density filter, Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved since it only affects the magnitude the intensity and not the polarization state According to (5-10), the emerging intensity I is then I ¼ p2 I ð5-11Þ where I is the intensity if the incident beam Equation (5-9) is the Mueller matrix for a polarizer which is described by unequal attenuations along the px and py axes An ideal linear polarizer is one which has transmission along only one axis and no transmission along the orthogonal axis This behavior can be described by first setting, say, py ¼ Then (5-9) reduces to 1 0 p2 B 1 0 C C ð5-12Þ M¼ xB @0 0 0A 0 0 Equation (5-12) is the Mueller matrix for an ideal linear polarizer which polarizes only along the x axis It is most often called a linear horizontal polarizer, arbitrarily assigning the horizontal to the x direction It would be a perfect linear polarizer if the transmission factor px was unity ð px ¼ 1Þ Thus, the Mueller matrix for an ideal perfect linear polarizer with its transmission axis in the x direction is 1 0 1B1 0C C ð5-13Þ M¼ B 2@0 0 0A 0 0 If the original beam is completely unpolarized, the maximum intensity of the emerging beam which can be obtained with a perfect ideal polarizer is only 50% of the original intensity It is the price we pay for obtaining perfectly polarized light If the original beam is perfectly horizontally polarized, there is no change in intensity This element is called a linear polarizer because it affects a linearly polarized beam in a unique manner as we shall soon see In general, all linear polarizers are described by (5-9) There is only one known natural material that comes close to approaching the perfect ideal polarizer described by (5-13), and this is calcite A synthetic material known as Polaroid is also used as a polarizer Its performance is not as good as calcite, but its cost is very low in comparison with that of natural calcite polarizers, e.g., a Glan–Thompson prism Nevertheless, there are a few types of Polaroid which perform extremely well as ‘‘ideal’’ polarizers We shall discuss the topic of calcite and Polaroid polarizers in Chapter 26 If an ideal perfect linear polarizer is used in which the role of the transmission axes is reversed from that of our linear horizontal polarizer, that is, px ¼ and py ¼ 1, then (5-9) reduces to 1 À1 0 B À1 0 C C ð5-14Þ M¼ B 0 0A 2@ 0 0 which is the Mueller matrix for a linear vertical polarizer Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Finally, it is convenient to rewrite the Mueller matrix, of a general linear polarizer, (5-9), in terms of trigonometric functions This can be done by setting p2x þ p2y ¼ p2 ð5-15aÞ and px ¼ p cos py ¼ p sin ð5-15bÞ Substituting (5-15) into (5-9) yields cos 2B cos p B M¼ B 2@ 0 sin 0 C C C A sin 0 ð5-16Þ where 90 For an ideal perfect linear polarizer p ¼ For a linear horizontal polarizer ¼ 0, and for a linear vertical polarizer ¼ 90 The usefulness of the trigonometric form of the Mueller matrix, (5-16), will appear later The reason for calling (5-13) and (5-14) linear polarizers is due to the following result Suppose we have an incident beam of arbitrary intensity and polarization so that its Stokes vector is S0 BS C B 1C S¼B C ð5-17Þ @ S2 A S3 We now matrix multiply (5-17) 0 01 Æ1 S0 B S C B Æ1 B B 1C B C¼ B @ S2 A @ 0 0 0 S3 by (5-13) or (5-14), and we can write 10 S0 BS C 0C CB C CB C A@ S2 A S3 Carrying out the matrix multiplication in (5-18), we find that 0 1 S0 B Æ1 C B S0 C B C B 1C C B C ¼ ðS0 Æ S1 ÞB @ A @ S2 A S 03 ð5-18Þ ð5-19Þ Inspecting (5-19), we see that the Stokes vector of the emerging beam is always linearly horizontally (þ) or vertically (À) polarized Thus an ideal linear polarizer always creates linearly polarized light regardless of the polarization state of the incident beam; however, note that because the factor 2px py in (5-9) is never zero, in practice there is no known perfect linear polarizer and all polarizers create elliptically polarized light While the ellipticity may be small and, in fact, negligible, there is always some present Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 5-3 Testing for a linear polarizer The above behavior of linear polarizers allows us to develop a test to determine if a polarizing element is actually a linear polarizer The test to determine if we have a linear polarizer is shown in Fig 5-3 In the test we assume that we have a linear polarizer and set its axis in the horizontal (H ) direction We then take another polarizer and set its axis in the vertical (V ) direction as shown in the figure The Stokes vector of the incident beam is S, and the Stokes vector of the beam emerging from the first polarizer (horizontal) is S ¼ MH S ð5-20Þ Next, the S beam propagates to the second polarizer (vertical), and the Stokes vector S 0 of the emerging beam is now S 0 ¼ MV S ¼ MV MH S ¼ MS ð5-21Þ where we have used (5-20) We see that M is the Mueller matrix of the combined vertical and linear polarizer: M ¼ MV M H ð5-22Þ where MH and MV are given by (5-13) and (5-14), respectively These results, (5-21) and (5-22), show that we can relate the Stokes vector of the emerging beam to the incident beam by merely multiplying the Mueller matrix of each component and finding the resulting Mueller matrix In general, the matrices not commute We now carry out the multiplication in (5-22) and write, using (5-13) and (5-14), 1 0 10 0 0 1 0 À1 0 C B C B 0C 1B B À1 CB 1 0 C B 0 0 C ð5-23Þ M¼ B C C¼B CB 4@ 0 0 A@ 0 0 A @ 0 0 A 0 0 0 0 0 0 Thus, we obtain a null Mueller matrix and, hence, a null output intensity regardless of the polarization state of the incident beam The appearance of a null Mueller matrix (or intensity) occurs only when the linear polarizers are in the crossed polarizer configuration Furthermore, the null Mueller matrix always arises whenever the polarizers are crossed, regardless of the angle of the transmission axis of the first polarizer Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 5.3 THE MUELLER MATRIX OF A RETARDER A retarder is a polarizing element which changes the phase of the optical beam Strictly speaking, its correct name is phase shifter However, historical usage has led to the alternative names retarder, wave plate, and compensator Retarders introduce a phase shift of between the orthogonal components of the incident field This can be thought of as being accomplished by causing a phase shift of þ =2 along the x axis and a phase shift of À =2 along the y axis These axes of the retarder are referred to as the fast and slow axes, respectively In Fig 5-4 we show the incident and emerging beam and the retarder The components of the emerging beam are related to the incident beam by E 0x ðz, tÞ ¼ eþi =2 Ex ðz, tÞ ð5-24aÞ E 0y ðz, tÞ ¼ eÀi =2 Ey ðz, tÞ ð5-24bÞ Referring again to the definition of the Stokes parameters (5-6) and (5-7) and substituting (5-24a) and (5-24b) into these equations, we find that S 00 ¼ S0 ð5-25aÞ S 01 ¼ S1 ð5-25bÞ S 02 ¼ S2 cos þ S3 sin ð5-25cÞ S 03 ¼ ÀS2 sin þ S3 cos ð5-25dÞ Equation (5-25) can 01 S0 B S 01 C B B C¼B @ S2 A @ 0 S 03 be written in matrix form as 10 0 S0 B S1 C 0 C CB C cos sin A@ S2 A À sin cos S3 ð5-26Þ Note that for an ideal phase shifter (retarder) there is no loss in intensity; that is, S 00 ¼ S0 Figure 5-4 Propagation of a polarized beam through a retarder Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved The Mueller matrix for a retarder with a phase shift is, from (5-26), 1 0 B0 0 C C M¼B ð5-27Þ @ 0 cos sin A 0 À sin cos There are two special cases of (5-27) which appear often in polarizing optics These are the cases for quarter-wave retarders ( ¼ 90 , i.e., the phase of one component of the light is delayed with respect to the orthogonal component by one quarter wave) and half-wave retarders ( ¼ 180 , i.e., the phase of one component of the light is delayed with respect to the orthogonal component by one half wave), respectively Obviously, a retarder is naturally dependent on wavelength, although there are achromatic retarders that are slowly dependent on wavelength We will discuss these topics in more detail in Chapter 26 For a quarter-wave retarder (5-27) becomes 1 0 B0 0C C M¼B ð5-28Þ @0 0 1A 0 À1 The quarter-wave retarder has the property that it transforms a linearly polarized beam with its axis at þ 45 or À 45 to the fast axis of the retarder into a right or left circularly polarized beam, respectively To show this property, consider the Stokes vector for a linearly polarized Æ 45 beam: 1 B 0C C S ¼ I0 B ð5-29Þ @ Æ1 A Multiplying (5-29) by (5-28) yields 1 B 0C C S ¼ I0 B @ 0A Ç1 ð5-30Þ which is the Stokes vector for left (right) circularly polarized light The transformation of linearly polarized light to circularly polarized light is an important application of quarter-wave retarders However, circularly polarized light is obtained only if the incident linearly polarized light is oriented at Æ 45 On the other hand, if the incident light is right (left) circularly polarized light, then multiplying (5-30) by (5-28) yields 1 B 0C C S ¼ I0 B ð5-31Þ @ Ç1 A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved which is the Stokes vector for linear À 45 or þ 45 polarized light The quarter-wave retarder can be used to transform linearly polarized light to circularly polarized light or circularly polarized light to linearly polarized light The other important type of wave retarder is the half-wave retarder ð ¼ 180 Þ For this condition (5-27) reduces to 1 0 B0 0C C M¼B ð5-32Þ @ 0 À1 0A 0 À1 A half-wave retarder is characterized by a diagonal matrix The terms m22 ¼ m33 ¼ À reverse the ellipticity and orientation of the polarization state of the incident beam To show this formally, we have initially S0 B S1 C C ð5-17Þ S¼B @ S2 A S3 We also saw previously that the orientation angle given in terms of the Stokes parameters: S2 S1 ð4-12Þ S3 S0 ð4-14Þ tan ¼ sin 2 ¼ and the ellipticity angle are Multiplying (5-17) by (5-32) gives 01 S0 S0 B S 01 C B S1 C C B C S0 ¼ B @ S 02 A ¼ @ ÀS2 A ÀS3 S 03 ð5-33Þ where tan S 02 S 01 ð5-34aÞ S 03 S 00 ð5-34bÞ ¼ sin 2 ¼ Substituting (5-33) into (5-34) yields tan ÀS2 ¼ À tan S1 ð5-35aÞ ÀS3 ¼ À sin 2 S0 ð5-35bÞ ¼ sin 2 ¼ Hence, 0 ¼ 90 À ¼ 90 þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-36aÞ ð5-36bÞ Half-wave retarders also possess the property that they can rotate the polarization ellipse This important property shall be discussed in Section 5.5 5.4 THE MUELLER MATRIX OF A ROTATOR The final way to change the polarization state of an optical field is to allow a beam to propagate through a polarizing element that rotates the orthogonal field components Ex(z, t) and Ey(z, t) through an angle In order to derive the Mueller matrix for rotation, we consider Fig 5-5 The angle describes the rotation of Ex to E 0x and of Ey to E 0y Similarly, the angle is the angle between E and Ex In the figure the point P is described in the E 0x , E 0y coordinate system by E 0x ¼ E cosð À Þ ð5-37aÞ E 0y ¼ E sinð À Þ ð5-37bÞ In the Ex, Ey coordinate system we have Ex ¼ E cos ð5-38aÞ Ey ¼ E sin ð5-38bÞ Expanding the trigonometric functions in (5-37) gives E 0x ¼ Eðcos cos þ sin sin Þ ð5-39aÞ E 0y ð5-39bÞ ¼ Eðsin cos À sin cos Þ Collecting terms in (5-39) using (5-38) then gives E 0x ¼ Ex cos þ Ey sin ð5-40aÞ E 0y ¼ ÀEx sin þ Ey cos ð5-40bÞ Figure 5-5 Rotation of the optical field components by a rotator Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Equations (5-40a) and (5-40b) are the amplitude equations for rotation In order to find the Mueller matrix we form the Stokes parameters for (5-40) as before and find the Mueller matrix for rotation: 1 0 B cos 2 sin 2 C C Mð2Þ ¼ B ð5-41Þ @ À sin 2 cos 2 A 0 We note that a physical rotation of leads to the appearance of 2 in (5-41) rather than because we are working in the intensity domain; in the amplitude domain we would expect just Rotators are primarily used to change the orientation angle of the polarization ellipse To see this behavior, suppose the orientation angle of an incident beam is Recall that tan ¼ S2 S1 ð4-12Þ For the emerging beam we have a similar expression with the variables in (4-12) replaced with primed variables Using (5-41) we see that the orientation angle is then tan ¼ ÀS1 sin 2 þ S2 cos 2 S1 cos 2 þ S2 sin 2 ð5-42Þ Equation (4-12) is now written as S2 ¼ S1 tan ð5-43Þ Substituting (5-43) into (5-42), we readily find that tan ¼ tanð2 À 2Þ ð5-44Þ À ð5-45Þ so ¼ Equation (5-45) shows that a rotator merely rotates the polarization ellipse of the incident beam; the ellipticity remains unchanged The sign is negative in (5-45) because the rotation is clockwise If the rotation is counterclockwise, that is, is replaced by À in (5-41), then we find ¼ þ ð5-46Þ In the derivation of the Mueller matrices for a polarizer, retarder, and rotator, we have assumed that the axes of these devices are aligned along the Ex and Ey (or x, y axes), respectively In practice, we find that the polarization elements are often rotated Consequently, it is also necessary for us to know the form of the Mueller matrices for the rotated polarizing elements We now consider this problem Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 5.5 MUELLER MATRICES FOR ROTATED POLARIZING COMPONENTS To derive the Mueller matrix for rotated polarizing components, we refer to Fig 5-6 The axes of the polarizing component are seen to be rotated through an angle to the x0 and y0 axes We must, therefore, also consider the components of the incident beam along the x0 and y0 axes In terms of the Stokes vector of the incident beam, S, we then have S ¼ MR ð2ÞS ð5-47Þ where MR(2) is the Mueller matrix for rotation (5-41) and S is the Stokes vector of the beam whose axes are along x0 and y0 The S beam now interacts with the polarizing element characterized by its Mueller matrix M The Stokes vector S00 of the beam emerging from the rotated polarizing component is S 0 ¼ MS ¼ MMR ð2ÞS ð5-48Þ where we have used (5-47) Finally, we must take the components of the emerging beam along the original x and y axes as seen in Fig 5-6 This can be described by a counterclockwise rotation of S00 through À and back to the original x, y axes, so S 0 ¼ MR ðÀ2ÞS 0 ¼ ½MR ðÀ2ÞMMR ð2ÞS ð5-49Þ where MR(À2) is, again, the Mueller matrix for rotation and S 000 is the Stokes vector of the emerging beam Equation (5-49) can be written as S 0 ¼ Mð2ÞS ð5-50Þ where Mð2Þ ¼ MR ðÀ2ÞMMR ð2Þ Figure 5-6 Derivation of the Mueller matrix for rotated polarizing components Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-51Þ Equation (5-51) is the Mueller matrix of a rotated polarizing component We recall that the Mueller matrix for rotation MR(2) is given by 1 0 B cos 2 sin 2 C C MR ð2Þ ¼ B ð5-52Þ @ À sin 2 cos 2 A 0 The rotated Mueller matrix expressed by (5-51) appears often in the treatment of polarized light Of particular interest are the Mueller matrices for a rotated polarizer and a rotated retarder The Mueller matrix for a rotated ‘‘rotator’’ is also interesting, but in a different way We recall that a rotator rotates the polarization ellipse by an amount If the rotator is now rotated through an angle , then one discovers, using (5-51), that M(2) ¼ MR(2); that is, the rotator is unaffected by a mechanical rotation Thus, the polarization ellipse cannot be rotated by rotating a rotator! The rotation comes about only by the intrinsic behavior of the rotator It is possible, however, to rotate the polarization ellipse mechanically by rotating a half-wave plate, as we shall soon demonstrate The Mueller matrix for a rotated polarizer is most conveniently found by expressing the Mueller matrix of a polarizer in angular form, namely, 1 cos 0 2B p cos 0 C C M¼ B ð5-16Þ @ 0 sin A 0 sin Carrying out the matrix multiplication according to (5-51) and using (5-52), the Mueller matrix for a rotated polarizer is 1 cos cos 2 cos sin 2 2 C 1B B cos cos 2 cos 2 þ sin sin 2 ð1 À sin 2 Þ sin 2 cos 2 C M¼ B C @ cos sin 2 ð1 À sin 2 Þ sin 2 cos 2 sin2 2 þ sin cos2 2 A 0 sin ð5-53Þ In (5-53) we have set p2 to unity We note that ¼ 0 , 45 , and 90 correspond to a linear horizontal polarizer, a neutral density filter, and a linear vertical polarizer, respectively The most common form of (5-53) is the Mueller matrix for an ideal linear horizontal polarizer ( ¼ 0 ) For this value (5-53) reduces to 1 cos 2 sin 2 sin 2 cos 2 C cos2 2 1B B cos 2 C MP ð2Þ ¼ B ð5-54Þ C @ sin 2 sin 2 cos 2 sin2 2 0A 0 0 In (5-54) we have written MP(2) to indicate that this is the Mueller matrix for a rotated ideal linear polarizer The form of (5-54) can be checked immediately by Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved setting ¼ (no rotation) Upon doing this, we obtain the Mueller matrix of a linear horizontal polarizer: 1B B MP ð0 Þ ¼ @ 0 1 0 0 0 0C C 0A ð5-55Þ One can readily see that for ¼ 45 and 90 (5-54) reduces to the Mueller matrix for an ideal linear þ 45 and vertical polarizer, respectively The Mueller matrix for a rotated ideal linear polarizer, (5-54), appears often in the generation and analysis of polarized light Next, we turn to determining the Mueller matrix for a retarder or wave plate We recall that the Mueller matrix for a retarder with phase shift is given by B0 Mc ¼ B @0 0 0 C C cos sin A À sin cos 0 ð5-56Þ Somtimes the term compensator is used in place of retarder, and so we have used the subscript ‘‘c.’’ From (5-51) the Mueller matrix for the rotated retarder (5-56) is found to be 1 0 B cos2 2 þ cos sin2 2 ð1 À cos Þ sin 2 cos 2 À sin sin 2 C B C Mc ð , 2Þ ¼ B C @ ð1 À cos Þ sin 2 cos 2 sin2 2 þ cos cos2 2 sin cos 2 A sin sin 2 À sin cos 2 cos ð5-57Þ For ¼ 0 , (5-57) reduces to (5-56) as expected There is a particularly interesting form of (5-57) for a phase shift of ¼ 180 , a so-called half-wave retarder For ¼ 180 (5-57) reduces to B cos 4 Mc ð180 , 4Þ ¼ B @ sin 4 0 sin 4 À cos 4 0 C C A À1 ð5-58Þ Equation (5-58) looks very similar to the Mueller matrix for rotation MR(2), (5-52), which we write simply as MR: B1 MR ¼ B @0 0 cos 2 À sin 2 0 sin 2 cos 2 0C C 0A Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-59Þ However, (5-58) differs from (5-59) in some essential ways The first is the ellipticity The Stokes vector of an incident beam is, as usual, S0 B C B S1 C C ð5-17Þ S¼B BS C @ 2A S3 Multiplying (5-17) by (5-59) yields the Stokes vector S : S0 B C B S1 cos 2 þ S2 sin 2 C C S0 ¼ B B ÀS sin 2 þ S cos 2 C @ A ð5-60Þ S3 The ellipticity angle 0 is sin 2 ¼ S 03 S3 ¼ ¼ sin 2 S 00 S0 ð5-61Þ Thus, the ellipticity is not changed under true rotation Multiplying (5-17) by (5-58), however, yields a Stokes vector S resulting from a half-wave retarder: S0 B C B S1 cos 4 þ S2 sin 4 C B C S ¼B ð5-62Þ C @ S1 sin 4 À S2 cos 4 A ÀS3 The ellipticity angle 0 is now sin 2 ¼ S 03 ÀS3 ¼ ¼ À sin 2 S 00 S0 ð5-63Þ Thus, ¼ þ 90 ð5-64Þ so the ellipticity angle of the incident beam is advanced 90 by using a rotated half-wave retarder The next difference is for the orientation angle For a rotator, (5-59), the orientation angle associated with the incident beam, , is given by the equation: tan ¼ S2 S1 ð5-65Þ so we immediately find from (5-65) and (5-60) that tan ¼ S 02 sin cos 2 À sin 2 cos sinð2 À 2Þ ¼ ¼ S 01 cos cos 2 þ sin sin 2 cosð2 À 2Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-66Þ whence À ¼ ð5-67Þ Equation (5-67) shows that a mechanical rotation in increases by the same amount and in the same direction (by definition, a clockwise rotation of increases) On the other hand, for a half-wave retarder the orientation angle is given by the equation, using (5-17) and (5-62), tan ¼ cos sin 4 À sin cos 4 sinð4 À Þ ¼ cos cos 4 þ sin sin 4 cosð4 À Þ ð5-68Þ so ¼ 2 À ð5-69aÞ ¼ Àð À 2Þ ð5-69bÞ or Comparing (5-69b) with (5-67), we see that rotating the half-wave retarder clockwise causes to rotate counterclockwise by an amount twice that of a rotator Because the rotation of a half-wave retarder is opposite to a true rotator, it is called a pseudorotator When a mechanical rotation of is made using a half-wave retarder the polarization ellipse is rotated by 2 and in a direction opposite to the direction of the mechanical rotation For a true mechanical rotation of the polarization ellipse is rotated by an amount and in the same direction as the rotation This discussion of rotation of half-wave retarders is more than academic, however Very often manufacturers sell half-wave retarders as polarization rotators Strictly speaking, this belief is quite correct However, one must realize that the use of a half-wave retarder rather than a true rotator requires a mechanical mount with twice the resolution That is, if we use a rotator in a mount with, say 20 of resolution, then in order to obtain the same resolution with a half-wave retarder a mechanical mount with 10 of resolution is required The simple fact is that doubling the resolution of a mechanical mount can be very expensive in comparison with using a true rotator The cost for doubling the resolution of a mechanical mount can easily double, whereas the cost increase between a quartz rotator and a half-wave retarder is usually much less In general, if the objective is to rotate the polarization ellipse by a known fixed amount, it is better to use a rotator rather than a half-wave retarder A half-wave retarder is very useful as a rotator Half-wave retarders can also be used to ‘‘reverse’’ the polarization state In order to illustrate this behavior, consider that we have an incident beam which is right or left circularly polarized Its Stokes vector is 1 B C C S ¼ I0 B ð5-70Þ @ A Æ1 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Multiplying (5-70) by (5-58) and setting ¼ 0 yields 1 B C C S ¼ I0 B @ A Ç1 ð5-71Þ We see that we again obtain circularly polarized light but opposite to its original state; that is, right circularly polarized light is transformed to left circularly polarized light, and vice versa Similarly, if we have incident linear þ 45 polarized light, the emerging beam is linear À 45 polarized light It is this property of reversing the ellipticity and the orientation, manifested by the negative sign in m22 and m33, that also makes half-wave plates very useful Finally, we consider the Mueller matrix of a rotated quarter-wave retarder We set ¼ 90 in (5-58) and we have B0 B Mc ð90 , 2Þ ¼ B @0 0 cos2 2 sin 2 cos 2 sin 2 cos 2 sin2 2 sin 2 À cos 2 0 À sin 2 C C C cos 2 A ð5-72Þ Consider that we have an incident linearly horizontally polarized beam, so its Stokes vector is ðI0 ¼ 1Þ 1 B1C C S¼B @0A ð5-73Þ We multiply (5-73) by (5-72), and we find that the Stokes vector S is 1 B cos2 2 C B C S0 ¼ B C @ sin 2 cos 2 A ð5-74Þ sin 2 We see immediately from (5-74) that the orientation angle angle of the emerging beam are given by tan ¼ tan 2 sin 2 ¼ sin 2 and the ellipticity ð5-75aÞ ð5-75bÞ Thus, the rotated quarter-wave plate has the property that it can be used to generate any desired orientation and ellipticity starting with an incident linearly horizontally polarized beam However, we can only select one of these parameters; we have no control over the other parameter We also note that if we initially have Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved right or left circularly polarized light the Stokes vector of the output beam is 1 B Ç sin 2 C C S0 ¼ B ð5-76Þ @ Æ cos 2 A which is the Stokes vector for linearly polarized light While it is well known that a quarter-wave retarder can be used to create linearly polarized light, (5-76) shows that an additional variation is possible by rotating the retarder, namely, the orientation can be controlled Equation (5-76) shows that we can generate any desired orientation or ellipticity of a beam, but not both This leads to the question of how we can generate an elliptically polarized beam of any desired orientation and ellipticity regardless of the polarization state of an incident beam 5.6 GENERATION OF ELLIPTICALLY POLARIZED LIGHT In the previous section we derived the Mueller matrices for a rotated polarizer and a rotated retarder We now apply these matrices to the generation of an elliptically polarized beam of any desired orientation and ellipticity In order to this we refer to Fig 5-7 In the figure we show an incident beam of arbitrary polarization The beam propagates first through an ideal polarizer rotated through an angle and then through a retarder, with its fast axis along the x axis The Stokes vector of the incident beam is S0 BS C B 1C S¼B C ð5-17Þ @ S2 A S3 Figure 5-7 The generation of elliptically polarized light Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved It is important that we consider the optical source to be arbitrarily polarized At first sight, for example, we might wish to use unpolarized light or linearly polarized light However, unpolarized light is surprisingly difficult to generate, and the requirement to generate ideal linearly polarized light calls for an excellent linear polarizer We can avoid this problem if we consider that the incident beam is of unknown but arbitrary polarization Our objective is to create an elliptically polarized beam of any desired ellipticity and orientation and which is totally independent of the polarization state of the incident beam The Mueller matrix of a rotated ideal linear polarizer is 1 cos 2 sin 2 cos2 2 sin 2 cos 2 C 1B B cos 2 C MP ð2Þ ¼ B ð5-54Þ C @ sin 2 sin 2 cos 2 sin2 2 0A 0 0 Multiplying (5-17) by (5-54) yields 1 B cos 2 C C S ¼ ðS0 þ S1 cos 2 þ S2 sin 2ÞB @ sin 2 A The Mueller matrix of the retarder (nonrotated) is 1 0 B0 0 C C Mc ¼ B @ 0 cos sin A 0 À sin cos ð5-77Þ ð5-56Þ Multiplying (5-77) by (5-56) then gives the Stokes vector of the beam emerging from the retarder: 1 B C cos 2 B C ð5-78aÞ S 0 ¼ IðÞB C @ cos sin 2 A À sin sin 2 where IðÞ ¼ ðS0 þ S1 cos 2 þ S2 sin 2Þ ð5-78bÞ Equation (5-78a) is the Stokes vector of an elliptically polarized beam We immediately find from (5-78a) that the orientation angle (we drop the double prime) is tan ¼ cos tan 2 ð5-79aÞ and the ellipticity angle is sin 2 ¼ À sin sin 2 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð5-79bÞ We must now determine the and which will generate the desired values of and We divide (5-79a) by tan 2 and (5-79b) by sin 2, square the equations, and add The result is cos 2 ¼ Æ cos 2 cos ð5-80Þ To determine the required phase shift , we divide (5-79b) by (5-79a): sin 2 ¼ À tan cos 2 tan ð5-81Þ Solving for tan and using (5-80), we easily find that tan ¼ Ç tan 2 sin ð5-82Þ Thus, (5-80) and (5-82) are the equations for the angles and to which the polarizer and the retarder must be set in order to obtain the desired ellipticity and orientation angles and We have thus shown that using only a rotated ideal linear polarizer and a retarder we can generate any state of elliptically polarized light There is a final interesting fact about (5-80) and (5-82) We write (5-80) and (5-82) as a pair in the form cos 2 ¼ Æ cos 2 cos ð5-80Þ tan 2 ¼ Ç sin tan ð5-83Þ Equations (5-80) and (5-83) are recognized as equations arising from spherical trigonometry for a right spherical triangle In Fig 5-8 we have drawn a right spherical triangle The angle (the orientation of the polarization ellipse) is plotted on the equator, and the angle 2 (the ellipticity of the polarization ellipse) is plotted on the longitude If a great circle is drawn from point A to point B, the length of the arc AB is given by (5-80) and corresponds to 2 as shown in the figure Similarly, the Figure 5-8 A right spherical triangle drawn on the surface of a sphere Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved phase is the angle between the arc AB and the equator; its value is given by (5-83) We see from Fig 5-8 that we can easily determine and by (1) measuring the length of the arc AB and (2) measuring the angle between the arc AB and the equator on a sphere The polarization equations (5-80) and (5-83) are intimately associated with spherical trigonometry and a sphere Furthermore, we recall from Section 4.3 that when the Stokes parameters were expressed in terms of the orientation angle and the ellipticity angle they led directly to the Poincare´ sphere In fact, (5-80) and (5-83) describe a spherical triangle which plots directly on to the Poincare´ sphere Thus, we see that even at this early stage in our study of polarized light there is a strong connection between the equations of polarized light and its representation on a sphere In fact, one of the most remarkable properties of polarized light is that there is such a close relation between these equations and the equations of spherical trigonometry In Chapter 12, on the Poincare´ sphere, these relations will be discussed in depth In order to provide the reader with background material on right spherical triangles a brief discussion of the fundamentals of spherical trigonometry is presented at the end of Section 12.2 REFERENCES Papers Soleillet, P., Ann Phys., 12 (10), 23 (1929) Perrin, F., J Chem Phys., 10, 415 (1942) Mueller, H., J Opt Soc Am., 37, 110 (1947) Parke, N G., III, Statistical Optics II: Mueller Phenomenological Algebra, RLE TR-119, Research Laboratory of Elect at M.I.T (1949) McMaster, W H., Am J Phys., 22, 351 (1954) Walker, M J., J Phys., 22, 170 (1954) McMaster, W H., Rev Mod Phys., 33, (1961) Collett, E., Am J Phys., 36, 713 (1968) Collett, E., Am J Phys., 39, 517 (1971) Books Shurcliff, W A., Polarized Light, Harvard University Press, Cambridge, MA, 1962 Gerrard, A and Burch, J M., Introduction to Matrix Methods in Optics, Wiley, London, 1975 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved [...]... Reserved 5. 5 MUELLER MATRICES FOR ROTATED POLARIZING COMPONENTS To derive the Mueller matrix for rotated polarizing components, we refer to Fig 5- 6 The axes of the polarizing component are seen to be rotated through an angle to the x0 and y0 axes We must, therefore, also consider the components of the incident beam along the x0 and y0 axes In terms of the Stokes vector of the incident beam, S, we then... 0A 0 5- 55 One can readily see that for ¼ 45 and 90 (5- 54) reduces to the Mueller matrix for an ideal linear þ 45 and vertical polarizer, respectively The Mueller matrix for a rotated ideal linear polarizer, (5- 54), appears often in the generation and analysis of polarized light Next, we turn to determining the Mueller matrix for a retarder or wave plate We recall that the Mueller matrix for a... 2 cos 5- 57Þ For ¼ 0 , (5- 57) reduces to (5- 56) as expected There is a particularly interesting form of (5- 57) for a phase shift of ¼ 180 , a so-called half-wave retarder For ¼ 180 (5- 57) reduces to 0 1 0 B 0 cos 4 Mc ð180 , 4Þ ¼ B @ 0 sin 4 0 0 0 sin 4 À cos 4 0 1 0 0 C C 0 A À1 5- 58Þ Equation (5- 58) looks very similar to the Mueller matrix for rotation MR(2), (5- 52), which we write... ðÀ2ÞMMR ð2Þ Figure 5- 6 Derivation of the Mueller matrix for rotated polarizing components Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 5- 51Þ Equation (5- 51) is the Mueller matrix of a rotated polarizing component We recall that the Mueller matrix for rotation MR(2) is given by 0 1 1 0 0 0 B 0 cos 2 sin 2 0 C C MR ð2Þ ¼ B 5- 52Þ @ 0 À sin 2 cos 2 0 A 0 0 0 1 The rotated Mueller matrix... ð2ÞS 5- 47Þ where MR(2) is the Mueller matrix for rotation (5- 41) and S 0 is the Stokes vector of the beam whose axes are along x0 and y0 The S 0 beam now interacts with the polarizing element characterized by its Mueller matrix M The Stokes vector S00 of the beam emerging from the rotated polarizing component is S 0 0 ¼ MS 0 ¼ MMR ð2ÞS 5- 48Þ where we have used (5- 47) Finally, we must take the components. .. then we find 0 ¼ þ 5- 46Þ In the derivation of the Mueller matrices for a polarizer, retarder, and rotator, we have assumed that the axes of these devices are aligned along the Ex and Ey (or x, y axes), respectively In practice, we find that the polarization elements are often rotated Consequently, it is also necessary for us to know the form of the Mueller matrices for the rotated polarizing elements... S1 tan 2 5- 43Þ Substituting (5- 43) into (5- 42), we readily find that tan 2 0 ¼ tanð2 À 2Þ 5- 44Þ À 5- 45 so 0 ¼ Equation (5- 45) shows that a rotator merely rotates the polarization ellipse of the incident beam; the ellipticity remains unchanged The sign is negative in (5- 45) because the rotation is clockwise If the rotation is counterclockwise, that is, is replaced by À in (5- 41), then we find... MP ð2Þ ¼ B 5- 54Þ C 2 @ sin 2 sin 2 cos 2 sin2 2 0A 0 0 0 0 Multiplying (5- 17) by (5- 54) yields 0 1 1 B cos 2 C 1 C S 0 ¼ ðS0 þ S1 cos 2 þ S2 sin 2ÞB @ sin 2 A 2 0 The Mueller matrix of the retarder (nonrotated) is 0 1 1 0 0 0 B0 1 0 0 C C Mc ¼ B @ 0 0 cos sin A 0 0 À sin cos 5- 77Þ 5- 56Þ Multiplying (5- 77) by (5- 56) then gives the Stokes vector of the beam emerging from the retarder:... write (5- 80) and (5- 82) as a pair in the form cos 2 ¼ Æ cos 2 cos 2 5- 80Þ tan 2 ¼ Ç sin 2 tan 5- 83Þ Equations (5- 80) and (5- 83) are recognized as equations arising from spherical trigonometry for a right spherical triangle In Fig 5- 8 we have drawn a right spherical triangle The angle 2 (the orientation of the polarization ellipse) is plotted on the equator, and the angle 2 (the ellipticity of the. .. Mueller matrix for rotation, we consider Fig 5- 5 The angle describes the rotation of Ex to E 0x and of Ey to E 0y Similarly, the angle is the angle between E and Ex In the figure the point P is described in the E 0x , E 0y coordinate system by E 0x ¼ E cosð À Þ 5- 37aÞ E 0y ¼ E sinð À Þ 5- 37bÞ In the Ex, Ey coordinate system we have Ex ¼ E cos 5- 38aÞ Ey ¼ E sin 5- 38bÞ Expanding the trigonometric